Total Value of an Constant Income Stream for a Fixed Time Interval
Suppose that USH Retail Incorporated receives income at a constant rate of 26000 dollars per year, then the total income received, T, from time t = 0 to t = 2 years is T = $26000* ( 2-0) = $52000.
Notice that at this annual rate USH receives $26000
/year * 1 year/ 52 weeks = $500 / week.
During the two years USH plans to deposit its income as
it is received in a savings account paying 3% interest per annum
, compounded continuously. Following this plan, the amount of money in
the account at the end of the two years, the future value (FV) of this
income, when time t = 2 can be estimated by looking at how
the weekly receipts accumulate once deposited in the savings account.
FV = 500 for 103 weeks of interest +500 for 102 weeks
of interest + 500 for 101 weeks of interest + ... +500 for 1 week of interest +500 for the last week ( no interest!)
= 500 *exp((.03)*103/52) +500 *exp((.03)*102/52)+500
*exp((.03)*101/52)+ ... +500 *exp((.03)*1/52 + 500)
Step | t = Part of Year | Weekly Amount Received | Contribution to Future Value |
1 | 1/52 | 500 | 500 *exp((.03)*103/52)) = 530.61 |
2 | 2/52 | 500 | 500 *exp((.03)*102/52)) = 530.30 |
3 | 3/52 | 500 | 500 *exp((.03)*101/52)) = 530.00 |
.
. . |
.
. . |
.
. . |
.
. . |
103 | 103/52 | 500 | 500 *exp((.03)*1/52)) = 500.28 |
104 |
104/52 |
500 |
500 |
sum | 52000 | $53576.22 |
Future value = FV = | ∫ | b
a |
Re^{.03(b-t)} dt = R/.03 [e^{.03(b-a)} -1] . |
Future value = FV = | ∫ | 2
0 |
26000e^{.03(2-t)} dt = 26000/.03 [e^{.03*2} -1] = 53591.67. |
Note: This Future Value is larger than the value computed in the table. This can be explained by the fact that in the continuous model for the computation, the deposits are decreasing continuously rather than discretely and therefore the table will give an underestimate for the continuous, or conversely the continuous model will give an overestimate compared to the discrete model.
Future value = FV = | ∫ | b
a |
Re^{r(b-t)} dt = R/r [e^{r(b-a)} -1] . |
If we consider the further generalization that the revenues rate of receipt varies so that R is a function of t, or R = R(t), then we have the final general formula
for future value:
Future value = FV = | ∫ | b
a |
R(t)e^{r(b-t)} dt. |
Suppose Ush Retail is up for sale. And the present rate of receipt of income is expected to continue at $26000 per year for the next two years and the income could be deposited as it is received in an account paying interest 3% per annum, compounded continuously. How much would we have to deposit now in the bank to achieve the same bank balance at the end of two years?
Solution: Let PV represent the amount would we have to deposit now in the bank to achieve the same bank balance at the end of two years. Then PV*e^{.03*2} is the amount that the Present Value would yield after two years in the bank. But this is supposed to equal the future value of the USH receipts, so
PV e^{.03*2 }= FV = | ∫ | 2
0 |
26000e^{.03(2-t)} dt. |
PV = e ^{-.03*2} | ∫ | 2
0 |
26000 e^{.03(2-t)} dt. |
PV =26000 | ∫ | 2
0 |
e ^{-.03t} dt = 26000/.03 [1-e^{-.03*2}]=$50470.73 . |
If the rate of receipt of income from time t = a to t = b is R dollars per unit of time and the income is deposited as it is received in an account paying interest r per unit of time, compounded continuously, then the present value of the income stream at time t = a is
Present value = PV = | ∫ | b
a |
Re^{r(a-t)} dt = R/r [1-e^{-r*b}]. |
Present value = PV = | ∫ | b
a |
R(t)e^{r(a-t)} dt. |
John Ush would like to retire in 30 years and thinks that this will be possible with ease if he has about $500,000 in savings. So he has decided to start a regular savings plan in an IRA saving account that he expects to continue paying 4% per year compounded continuously. How much should John plan to put into his account on a monthly basis to achieve his goal?
Solution: Since John wants to accumulate $500,000, we use this for the future value of the account and using the formula for FV we need to solve for R.
Future value = FV = 500,000 = | ∫ | 30
0 |
Re^{.04(30-t)} dt = R/.04 *[e^{.04(30)} -1] . |
Thus R = 500,000 (.04) /[e^{.04(30)} -1] = 20000/ 2.32 = 8620.25.
Discussion: If John can find a better interest rate for his savings, perhaps 6%, and he plans to retire in 40 years (he's now 25 years old) then he would need to save only about R = 500,000 (.06) /[e^{.06(40)} -1] = 30000/10.02 = 2993.06 annually or about $249.42 per month!